<!DOCTYPE html><html lang="en" data-theme="light"><head><meta charset="UTF-8"><meta http-equiv="X-UA-Compatible" content="IE=edge"><meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no"><title>数学分析1-史济怀-集合的映射，集合的势 | cuola</title><meta name="keywords" content="数学分析 集合"><meta name="author" content="cuola"><meta name="copyright" content="cuola"><meta name="format-detection" content="telephone=no"><meta name="theme-color" content="#ffffff"><meta name="description" content="limn→+∞∑i&#x3D;0n1i!&#x3D;elim_{n\rightarrow+\infty}\sum_{i&#x3D;0}^n\frac{1}{i!}&#x3D;elimn→+∞​∑i&#x3D;0n​i!1​&#x3D;e 交错级数 ∑n&#x3D;1∞(−1)n−11n&#x3D;ln2\sum_{n&#x3D;1}^{\infty}(-1)^{n-1}\frac{1}{n}&#x3D;ln2∑n&#x3D;1∞​(−1)n−1n1​&#x3D;ln2 证明： limn→+∞S2nlim_{n\ri">
<meta property="og:type" content="article">
<meta property="og:title" content="数学分析1-史济怀-集合的映射，集合的势">
<meta property="og:url" content="http://example.com/2022/08/28/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%901-%E5%8F%B2%E6%B5%8E%E6%80%80-%E9%9B%86%E5%90%88%E7%9A%84%E6%98%A0%E5%B0%84%EF%BC%8C%E9%9B%86%E5%90%88%E7%9A%84%E5%8A%BF/index.html">
<meta property="og:site_name" content="cuola">
<meta property="og:description" content="limn→+∞∑i&#x3D;0n1i!&#x3D;elim_{n\rightarrow+\infty}\sum_{i&#x3D;0}^n\frac{1}{i!}&#x3D;elimn→+∞​∑i&#x3D;0n​i!1​&#x3D;e 交错级数 ∑n&#x3D;1∞(−1)n−11n&#x3D;ln2\sum_{n&#x3D;1}^{\infty}(-1)^{n-1}\frac{1}{n}&#x3D;ln2∑n&#x3D;1∞​(−1)n−1n1​&#x3D;ln2 证明： limn→+∞S2nlim_{n\ri">
<meta property="og:locale" content="en_US">
<meta property="og:image" content="http://example.com/img/back1.jpg">
<meta property="article:published_time" content="2022-08-28T08:32:21.000Z">
<meta property="article:modified_time" content="2022-08-28T15:34:07.495Z">
<meta property="article:author" content="cuola">
<meta property="article:tag" content="数学分析 集合">
<meta name="twitter:card" content="summary">
<meta name="twitter:image" content="http://example.com/img/back1.jpg"><link rel="shortcut icon" href="/img/favicon.png"><link rel="canonical" href="http://example.com/2022/08/28/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%901-%E5%8F%B2%E6%B5%8E%E6%80%80-%E9%9B%86%E5%90%88%E7%9A%84%E6%98%A0%E5%B0%84%EF%BC%8C%E9%9B%86%E5%90%88%E7%9A%84%E5%8A%BF/"><link rel="preconnect" href="//cdn.jsdelivr.net"/><link rel="preconnect" href="//busuanzi.ibruce.info"/><link rel="stylesheet" href="/css/index.css"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/@fortawesome/fontawesome-free/css/all.min.css" media="print" onload="this.media='all'"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/@fancyapps/ui/dist/fancybox.min.css" media="print" onload="this.media='all'"><script>const GLOBAL_CONFIG = { 
  root: '/',
  algolia: undefined,
  localSearch: {"path":"/search.xml","preload":false,"languages":{"hits_empty":"We didn't find any results for the search: ${query}"}},
  translate: undefined,
  noticeOutdate: undefined,
  highlight: {"plugin":"highlighjs","highlightCopy":true,"highlightLang":true,"highlightHeightLimit":false},
  copy: {
    success: 'Copy successfully',
    error: 'Copy error',
    noSupport: 'The browser does not support'
  },
  relativeDate: {
    homepage: false,
    post: false
  },
  runtime: '',
  date_suffix: {
    just: 'Just',
    min: 'minutes ago',
    hour: 'hours ago',
    day: 'days ago',
    month: 'months ago'
  },
  copyright: undefined,
  lightbox: 'fancybox',
  Snackbar: undefined,
  source: {
    justifiedGallery: {
      js: 'https://cdn.jsdelivr.net/npm/flickr-justified-gallery/dist/fjGallery.min.js',
      css: 'https://cdn.jsdelivr.net/npm/flickr-justified-gallery/dist/fjGallery.min.css'
    }
  },
  isPhotoFigcaption: false,
  islazyload: false,
  isAnchor: false
}</script><script id="config-diff">var GLOBAL_CONFIG_SITE = {
  title: '数学分析1-史济怀-集合的映射，集合的势',
  isPost: true,
  isHome: false,
  isHighlightShrink: false,
  isToc: true,
  postUpdate: '2022-08-28 23:34:07'
}</script><noscript><style type="text/css">
  #nav {
    opacity: 1
  }
  .justified-gallery img {
    opacity: 1
  }

  #recent-posts time,
  #post-meta time {
    display: inline !important
  }
</style></noscript><script>(win=>{
    win.saveToLocal = {
      set: function setWithExpiry(key, value, ttl) {
        if (ttl === 0) return
        const now = new Date()
        const expiryDay = ttl * 86400000
        const item = {
          value: value,
          expiry: now.getTime() + expiryDay,
        }
        localStorage.setItem(key, JSON.stringify(item))
      },

      get: function getWithExpiry(key) {
        const itemStr = localStorage.getItem(key)

        if (!itemStr) {
          return undefined
        }
        const item = JSON.parse(itemStr)
        const now = new Date()

        if (now.getTime() > item.expiry) {
          localStorage.removeItem(key)
          return undefined
        }
        return item.value
      }
    }
  
    win.getScript = url => new Promise((resolve, reject) => {
      const script = document.createElement('script')
      script.src = url
      script.async = true
      script.onerror = reject
      script.onload = script.onreadystatechange = function() {
        const loadState = this.readyState
        if (loadState && loadState !== 'loaded' && loadState !== 'complete') return
        script.onload = script.onreadystatechange = null
        resolve()
      }
      document.head.appendChild(script)
    })
  
      win.activateDarkMode = function () {
        document.documentElement.setAttribute('data-theme', 'dark')
        if (document.querySelector('meta[name="theme-color"]') !== null) {
          document.querySelector('meta[name="theme-color"]').setAttribute('content', '#0d0d0d')
        }
      }
      win.activateLightMode = function () {
        document.documentElement.setAttribute('data-theme', 'light')
        if (document.querySelector('meta[name="theme-color"]') !== null) {
          document.querySelector('meta[name="theme-color"]').setAttribute('content', '#ffffff')
        }
      }
      const t = saveToLocal.get('theme')
    
          if (t === 'dark') activateDarkMode()
          else if (t === 'light') activateLightMode()
        
      const asideStatus = saveToLocal.get('aside-status')
      if (asideStatus !== undefined) {
        if (asideStatus === 'hide') {
          document.documentElement.classList.add('hide-aside')
        } else {
          document.documentElement.classList.remove('hide-aside')
        }
      }
    
    const detectApple = () => {
      if(/iPad|iPhone|iPod|Macintosh/.test(navigator.userAgent)){
        document.documentElement.classList.add('apple')
      }
    }
    detectApple()
    })(window)</script><link rel="stylesheet" href="/css/universe.css"><meta name="generator" content="Hexo 6.2.0"></head><body><div id="sidebar"><div id="menu-mask"></div><div id="sidebar-menus"><div class="avatar-img is-center"><img src="/img/favicon.png" onerror="onerror=null;src='/img/friend_404.gif'" alt="avatar"/></div><div class="sidebar-site-data site-data is-center"><a href="/archives/"><div class="headline">Articles</div><div class="length-num">15</div></a><a href="/tags/"><div class="headline">Tags</div><div class="length-num">3</div></a><a href="/categories/"><div class="headline">Categories</div><div class="length-num">1</div></a></div><hr/></div></div><div class="post" id="body-wrap"><header class="post-bg" id="page-header" style="background-image: url('/img/back1.jpg')"><nav id="nav"><span id="blog_name"><a id="site-name" href="/">cuola</a></span><div id="menus"><div id="search-button"><a class="site-page social-icon search"><i class="fas fa-search fa-fw"></i><span> Search</span></a></div><div id="toggle-menu"><a class="site-page"><i class="fas fa-bars fa-fw"></i></a></div></div></nav><div id="post-info"><h1 class="post-title">数学分析1-史济怀-集合的映射，集合的势</h1><div id="post-meta"><div class="meta-firstline"><span class="post-meta-date"><i class="far fa-calendar-alt fa-fw post-meta-icon"></i><span class="post-meta-label">Created</span><time class="post-meta-date-created" datetime="2022-08-28T08:32:21.000Z" title="Created 2022-08-28 16:32:21">2022-08-28</time><span class="post-meta-separator">|</span><i class="fas fa-history fa-fw post-meta-icon"></i><span class="post-meta-label">Updated</span><time class="post-meta-date-updated" datetime="2022-08-28T15:34:07.495Z" title="Updated 2022-08-28 23:34:07">2022-08-28</time></span><span class="post-meta-categories"><span class="post-meta-separator">|</span><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/categories/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%901/">数学分析1</a></span></div><div class="meta-secondline"><span class="post-meta-separator">|</span><span id="" data-flag-title="数学分析1-史济怀-集合的映射，集合的势"><i class="far fa-eye fa-fw post-meta-icon"></i><span class="post-meta-label">Post View:</span><span id="twikoo_visitors"><i class="fa-solid fa-spinner fa-spin"></i></span></span></div></div></div></header><main class="layout" id="content-inner"><div id="post"><article class="post-content" id="article-container"><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><msub><mi>m</mi><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msub><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>n</mi></msubsup><mfrac><mn>1</mn><mrow><mi>i</mi><mo stretchy="false">!</mo></mrow></mfrac><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">lim_{n\rightarrow+\infty}\sum_{i=0}^n\frac{1}{i!}=e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mclose mtight">!</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span></p>
<p>交错级数</p>
<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mfrac><mn>1</mn><mi>n</mi></mfrac><mo>=</mo><mi>l</mi><mi>n</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">\sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{n}=ln2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">n</span><span class="mord">2</span></span></span></span></p>
<p>证明：</p>
<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><msub><mi>m</mi><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msub><msub><mi>S</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">lim_{n\rightarrow+\infty}S_{2n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>=<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><msub><mi>m</mi><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msub><msub><mi>S</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">lim_{n\rightarrow+\infty}S_{2n+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>S</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mo>⋯</mo><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>+</mo><mo>⋯</mo><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>n</mi></mrow></mfrac><mo>−</mo><mn>2</mn><mo>∗</mo><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mo>⋯</mo><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>n</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mo>⋯</mo><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>n</mi></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
S_{2n} &amp;= 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots+\frac{1} {2n-1}-\frac{1}{2n} \\
       &amp;=1 + \frac{1}{2} + \frac{1}{3} + \cdots+\frac{1}{2n-1}+\frac{1}{2n}-2*(\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2n})\\
       &amp;=\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n-1}+\frac{1}{2n}\\
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.1723em;vertical-align:-3.3362em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.8362em;"><span style="top:-5.8362em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.4454em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span><span style="top:-1.0546em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.3362em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.8362em;"><span style="top:-5.8362em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.4454em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span><span style="top:-1.0546em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.3362em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><msub><mi>m</mi><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msub><msub><mi>S</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>=</mo><mi>l</mi><mi>n</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">lim_{n\rightarrow+\infty}S_{2n} = ln2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">n</span><span class="mord">2</span></span></span></span></p>
<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>S</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">S_{2n+1}=S_{2n}+\frac{1}{2n+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2484em;vertical-align:-0.4033em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></p>
<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><msub><mi>m</mi><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msub><msub><mi>S</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>l</mi><mi>n</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">lim_{n\rightarrow+\infty}S_{2n+1}=ln2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">n</span><span class="mord">2</span></span></span></span></p>
<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><msub><mi>m</mi><mrow><mi>n</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msub><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><mi>l</mi><mi>n</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">lim_{n\rightarrow+\infty}S_n=ln2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">→</span><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">n</span><span class="mord">2</span></span></span></span></p>
<h1>函数的连续性</h1>
<h2 id="函数的映射">函数的映射</h2>
<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\rightarrow B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 映射</p>
<p>满射：B中所有的元素，A中都有一个可以映射到它</p>
<p>单射：A中任意两个不同的元素，对应B中不同的元素</p>
<p>满射且单射：一一对应</p>
<h2 id="集合的势">集合的势</h2>
<p>有限集：可以比较元素数量的多少</p>
<p>无限集：无法知道具体数量</p>
<p>Bolzano</p>
<p>A~B:对等，等价(A和B有相同的势)</p>
<p>可数集：如果一个集合和自然数集对等，则是可数集</p>
<p>[0, 1)内的有理数是可数的</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>r</mi><mo>=</mo><mfrac><mi>p</mi><mi>q</mi></mfrac><mo stretchy="false">(</mo><mi>p</mi><mo>&lt;</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r=\frac{p}{q}(p&lt;q)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.988em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mclose">)</span></span></span></span></span></p>
<h3 id="定理：一个可数集的任何无限子集仍是可数集">定理：一个可数集的任何无限子集仍是可数集</h3>
<h3 id="定理：可数多可数集的并仍是可数集">定理：可数多可数集的并仍是可数集</h3>
<h3 id="定理：R中全体有理数是可数的">定理：R中全体有理数是可数的</h3>
<h3 id="定理：-0-1-中的实数是不可数的">定理：[0,1)中的实数是不可数的</h3>
<p>有理数都是代数数</p>
<p>无理数分为代数数和超越数</p>
<p>e和<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span> 都是超越数</p>
<h2 id="函数">函数</h2>
</article><div class="post-copyright"><div class="post-copyright__author"><span class="post-copyright-meta">Author: </span><span class="post-copyright-info"><a href="http://example.com">cuola</a></span></div><div class="post-copyright__type"><span class="post-copyright-meta">Link: </span><span class="post-copyright-info"><a href="http://example.com/2022/08/28/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%901-%E5%8F%B2%E6%B5%8E%E6%80%80-%E9%9B%86%E5%90%88%E7%9A%84%E6%98%A0%E5%B0%84%EF%BC%8C%E9%9B%86%E5%90%88%E7%9A%84%E5%8A%BF/">http://example.com/2022/08/28/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%901-%E5%8F%B2%E6%B5%8E%E6%80%80-%E9%9B%86%E5%90%88%E7%9A%84%E6%98%A0%E5%B0%84%EF%BC%8C%E9%9B%86%E5%90%88%E7%9A%84%E5%8A%BF/</a></span></div><div class="post-copyright__notice"><span class="post-copyright-meta">Copyright Notice: </span><span class="post-copyright-info">All articles in this blog are licensed under <a target="_blank" rel="noopener" href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a> unless stating additionally.</span></div></div><div class="tag_share"><div class="post-meta__tag-list"><a class="post-meta__tags" href="/tags/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90-%E9%9B%86%E5%90%88/">数学分析 集合</a></div><div class="post_share"><div class="social-share" data-image="/img/back1.jpg" data-sites="facebook,twitter,wechat,weibo,qq"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/butterfly-extsrc/sharejs/dist/css/share.min.css" media="print" onload="this.media='all'"><script src="https://cdn.jsdelivr.net/npm/butterfly-extsrc/sharejs/dist/js/social-share.min.js" defer></script></div></div><nav class="pagination-post" id="pagination"><div class="prev-post pull-left"><a href="/2022/08/29/test/"><img class="prev-cover" src="/img/back1.jpg" onerror="onerror=null;src='/img/404.jpg'" alt="cover of previous post"><div class="pagination-info"><div class="label">Previous Post</div><div class="prev_info">test</div></div></a></div><div class="next-post pull-right"><a href="/2022/08/23/Mathjax%E4%BD%BF%E7%94%A8%E6%8C%87%E5%8D%97(%E4%B8%8D%E6%96%AD%E6%9B%B4%E6%96%B0)/"><img class="next-cover" src="/img/ani.png" onerror="onerror=null;src='/img/404.jpg'" alt="cover of next post"><div class="pagination-info"><div class="label">Next Post</div><div class="next_info">Mathjax使用指南(不断更新)</div></div></a></div></nav><hr/><div id="post-comment"><div class="comment-head"><div class="comment-headline"><i class="fas fa-comments fa-fw"></i><span> Comment</span></div></div><div class="comment-wrap"><div><div id="twikoo-wrap"></div></div></div></div></div><div class="aside-content" id="aside-content"><div class="card-widget card-info"><div class="is-center"><div class="avatar-img"><img src="/img/favicon.png" onerror="this.onerror=null;this.src='/img/friend_404.gif'" alt="avatar"/></div><div class="author-info__name">cuola</div><div class="author-info__description">你的名字是我的所有</div></div><div class="card-info-data site-data is-center"><a href="/archives/"><div class="headline">Articles</div><div class="length-num">15</div></a><a href="/tags/"><div class="headline">Tags</div><div class="length-num">3</div></a><a href="/categories/"><div class="headline">Categories</div><div class="length-num">1</div></a></div><a id="card-info-btn" target="_blank" rel="noopener" href="https://github.com/cuola/cuola.github.io"><i class="fab fa-github"></i><span>Follow Me</span></a></div><div class="card-widget card-announcement"><div class="item-headline"><i class="fas fa-bullhorn fa-shake"></i><span>Announcement</span></div><div class="announcement_content">This is my Blog</div></div><div class="sticky_layout"><div class="card-widget" id="card-toc"><div class="item-headline"><i class="fas fa-stream"></i><span>Catalog</span><span class="toc-percentage"></span></div><div class="toc-content"><ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link"><span class="toc-number">1.</span> <span class="toc-text">函数的连续性</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%87%BD%E6%95%B0%E7%9A%84%E6%98%A0%E5%B0%84"><span class="toc-number">1.1.</span> <span class="toc-text">函数的映射</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E9%9B%86%E5%90%88%E7%9A%84%E5%8A%BF"><span class="toc-number">1.2.</span> <span class="toc-text">集合的势</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%AE%9A%E7%90%86%EF%BC%9A%E4%B8%80%E4%B8%AA%E5%8F%AF%E6%95%B0%E9%9B%86%E7%9A%84%E4%BB%BB%E4%BD%95%E6%97%A0%E9%99%90%E5%AD%90%E9%9B%86%E4%BB%8D%E6%98%AF%E5%8F%AF%E6%95%B0%E9%9B%86"><span class="toc-number">1.2.1.</span> <span class="toc-text">定理：一个可数集的任何无限子集仍是可数集</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%AE%9A%E7%90%86%EF%BC%9A%E5%8F%AF%E6%95%B0%E5%A4%9A%E5%8F%AF%E6%95%B0%E9%9B%86%E7%9A%84%E5%B9%B6%E4%BB%8D%E6%98%AF%E5%8F%AF%E6%95%B0%E9%9B%86"><span class="toc-number">1.2.2.</span> <span class="toc-text">定理：可数多可数集的并仍是可数集</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%AE%9A%E7%90%86%EF%BC%9AR%E4%B8%AD%E5%85%A8%E4%BD%93%E6%9C%89%E7%90%86%E6%95%B0%E6%98%AF%E5%8F%AF%E6%95%B0%E7%9A%84"><span class="toc-number">1.2.3.</span> <span class="toc-text">定理：R中全体有理数是可数的</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%AE%9A%E7%90%86%EF%BC%9A-0-1-%E4%B8%AD%E7%9A%84%E5%AE%9E%E6%95%B0%E6%98%AF%E4%B8%8D%E5%8F%AF%E6%95%B0%E7%9A%84"><span class="toc-number">1.2.4.</span> <span class="toc-text">定理：[0,1)中的实数是不可数的</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%87%BD%E6%95%B0"><span class="toc-number">1.3.</span> <span class="toc-text">函数</span></a></li></ol></li></ol></div></div><div class="card-widget card-recent-post"><div class="item-headline"><i class="fas fa-history"></i><span>Recent Post</span></div><div class="aside-list"><div class="aside-list-item"><a class="thumbnail" href="/2022/08/29/test2/" title="test2"><img src="/img/back1.jpg" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="test2"/></a><div class="content"><a class="title" href="/2022/08/29/test2/" title="test2">test2</a><time datetime="2022-08-29T04:20:30.000Z" title="Created 2022-08-29 12:20:30">2022-08-29</time></div></div><div class="aside-list-item"><a class="thumbnail" href="/2022/08/29/test/" title="test"><img src="/img/back1.jpg" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="test"/></a><div class="content"><a class="title" href="/2022/08/29/test/" title="test">test</a><time datetime="2022-08-28T16:26:36.000Z" title="Created 2022-08-29 00:26:36">2022-08-29</time></div></div><div class="aside-list-item"><a class="thumbnail" href="/2022/08/28/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%901-%E5%8F%B2%E6%B5%8E%E6%80%80-%E9%9B%86%E5%90%88%E7%9A%84%E6%98%A0%E5%B0%84%EF%BC%8C%E9%9B%86%E5%90%88%E7%9A%84%E5%8A%BF/" title="数学分析1-史济怀-集合的映射，集合的势"><img src="/img/back1.jpg" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="数学分析1-史济怀-集合的映射，集合的势"/></a><div class="content"><a class="title" href="/2022/08/28/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%901-%E5%8F%B2%E6%B5%8E%E6%80%80-%E9%9B%86%E5%90%88%E7%9A%84%E6%98%A0%E5%B0%84%EF%BC%8C%E9%9B%86%E5%90%88%E7%9A%84%E5%8A%BF/" title="数学分析1-史济怀-集合的映射，集合的势">数学分析1-史济怀-集合的映射，集合的势</a><time datetime="2022-08-28T08:32:21.000Z" title="Created 2022-08-28 16:32:21">2022-08-28</time></div></div><div class="aside-list-item"><a class="thumbnail" href="/2022/08/23/Mathjax%E4%BD%BF%E7%94%A8%E6%8C%87%E5%8D%97(%E4%B8%8D%E6%96%AD%E6%9B%B4%E6%96%B0)/" title="Mathjax使用指南(不断更新)"><img src="/img/ani.png" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="Mathjax使用指南(不断更新)"/></a><div class="content"><a class="title" href="/2022/08/23/Mathjax%E4%BD%BF%E7%94%A8%E6%8C%87%E5%8D%97(%E4%B8%8D%E6%96%AD%E6%9B%B4%E6%96%B0)/" title="Mathjax使用指南(不断更新)">Mathjax使用指南(不断更新)</a><time datetime="2022-08-23T08:27:02.000Z" title="Created 2022-08-23 16:27:02">2022-08-23</time></div></div><div class="aside-list-item"><a class="thumbnail" href="/2022/08/04/%E6%8A%B1%E6%AD%89%EF%BC%8C%E5%AE%8B%E5%A8%9C/" title="抱歉，宋娜"><img src="/img/4.png" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="抱歉，宋娜"/></a><div class="content"><a class="title" href="/2022/08/04/%E6%8A%B1%E6%AD%89%EF%BC%8C%E5%AE%8B%E5%A8%9C/" title="抱歉，宋娜">抱歉，宋娜</a><time datetime="2022-08-04T08:02:51.000Z" title="Created 2022-08-04 16:02:51">2022-08-04</time></div></div></div></div></div></div></main><footer id="footer" style="background-image: url('/img/foot.jpg')"><div id="footer-wrap"><div class="copyright">&copy;2020 - 2022 By cuola</div><div class="framework-info"><span>Framework </span><a target="_blank" rel="noopener" href="https://hexo.io">Hexo</a><span class="footer-separator">|</span><span>Theme </span><a target="_blank" rel="noopener" href="https://github.com/jerryc127/hexo-theme-butterfly">Butterfly</a></div></div></footer></div><div id="rightside"><div id="rightside-config-hide"><button id="readmode" type="button" title="Read Mode"><i class="fas fa-book-open"></i></button><button id="darkmode" type="button" title="Switch Between Light And Dark Mode"><i class="fas fa-adjust"></i></button><button id="hide-aside-btn" type="button" title="Toggle between single-column and double-column"><i class="fas fa-arrows-alt-h"></i></button></div><div id="rightside-config-show"><button id="rightside_config" type="button" title="Setting"><i class="fas fa-cog fa-spin"></i></button><button class="close" id="mobile-toc-button" type="button" title="Table Of Contents"><i class="fas fa-list-ul"></i></button><a id="to_comment" href="#post-comment" title="Scroll To Comments"><i class="fas fa-comments"></i></a><button id="go-up" type="button" title="Back To Top"><i class="fas fa-arrow-up"></i></button></div></div><div id="local-search"><div class="search-dialog"><nav class="search-nav"><span class="search-dialog-title">Search</span><span id="loading-status"></span><button class="search-close-button"><i class="fas fa-times"></i></button></nav><div class="is-center" id="loading-database"><i class="fas fa-spinner fa-pulse"></i><span>  Loading the Database</span></div><div class="search-wrap"><div id="local-search-input"><div class="local-search-box"><input class="local-search-box--input" placeholder="Search for Posts" type="text"/></div></div><hr/><div id="local-search-results"></div></div></div><div id="search-mask"></div></div><div><script src="/js/utils.js"></script><script src="/js/main.js"></script><script src="https://cdn.jsdelivr.net/npm/@fancyapps/ui/dist/fancybox.umd.min.js"></script><script src="/js/search/local-search.js"></script><div class="js-pjax"><script>(()=>{
  const init = () => {
    twikoo.init(Object.assign({
      el: '#twikoo-wrap',
      envId: 'https://twikoo-cuola.vercel.app/',
      region: '',
      onCommentLoaded: function () {
        btf.loadLightbox(document.querySelectorAll('#twikoo .tk-content img:not(.tk-owo-emotion)'))
      }
    }, null))
  }

  const getCount = () => {
    const countELement = document.getElementById('twikoo-count')
    if(!countELement) return
    twikoo.getCommentsCount({
      envId: 'https://twikoo-cuola.vercel.app/',
      region: '',
      urls: [window.location.pathname],
      includeReply: false
    }).then(function (res) {
      countELement.innerText = res[0].count
    }).catch(function (err) {
      console.error(err);
    });
  }

  const runFn = () => {
    init()
    
  }

  const loadTwikoo = () => {
    if (typeof twikoo === 'object') {
      setTimeout(runFn,0)
      return
    } 
    getScript('https://cdn.jsdelivr.net/npm/twikoo/dist/twikoo.all.min.js').then(runFn)
  }

  if ('Twikoo' === 'Twikoo' || !false) {
    if (false) btf.loadComment(document.getElementById('twikoo-wrap'), loadTwikoo)
    else loadTwikoo()
  } else {
    window.loadOtherComment = () => {
      loadTwikoo()
    }
  }
})()</script></div><div class="aplayer no-destroy" data-id="7605212209" data-server="netease" data-type="playlist" data-fixed="true" data-autoplay="true" data-lrcType="-1"> </div><canvas id="universe"></canvas><script defer src="/js/universe.js"></script><script src="https://cdn.jsdelivr.net/npm/butterfly-extsrc/dist/activate-power-mode.min.js"></script><script>POWERMODE.colorful = true;
POWERMODE.shake = true;
POWERMODE.mobile = false;
document.body.addEventListener('input', POWERMODE);
</script><script>window.$crisp = [];
window.CRISP_WEBSITE_ID = "3ec51170-6951-4ce7-a0a1-4abf9e0c9012";
(function () {
  d = document;
  s = d.createElement("script");
  s.src = "https://client.crisp.chat/l.js";
  s.async = 1;
  d.getElementsByTagName("head")[0].appendChild(s);
})();
$crisp.push(["safe", true])

if (false) {
  $crisp.push(["do", "chat:hide"])
  $crisp.push(["on", "chat:closed", function() {
    $crisp.push(["do", "chat:hide"])
  }])
  var chatBtnFn = () => {
    var chatBtn = document.getElementById("chat_btn")
    chatBtn.addEventListener("click", function(){
      $crisp.push(["do", "chat:show"])
      $crisp.push(["do", "chat:open"])

    });
  }
  chatBtnFn()
} else {
  if (false) {
    function chatBtnHide () {
      $crisp.push(["do", "chat:hide"])
    }
    function chatBtnShow () {
      $crisp.push(["do", "chat:show"])
    }
  }
}</script><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/aplayer/dist/APlayer.min.css" media="print" onload="this.media='all'"><script src="https://cdn.jsdelivr.net/npm/aplayer/dist/APlayer.min.js"></script><script src="https://cdn.jsdelivr.net/npm/butterfly-extsrc/metingjs/dist/Meting.min.js"></script><script src="https://cdn.jsdelivr.net/npm/pjax/pjax.min.js"></script><script>let pjaxSelectors = ["head > title","#config-diff","#body-wrap","#rightside-config-hide","#rightside-config-show",".js-pjax"]

var pjax = new Pjax({
  elements: 'a:not([target="_blank"])',
  selectors: pjaxSelectors,
  cacheBust: false,
  analytics: false,
  scrollRestoration: false
})

document.addEventListener('pjax:send', function () {

  // removeEventListener scroll 
  window.tocScrollFn && window.removeEventListener('scroll', window.tocScrollFn)
  window.scrollCollect && window.removeEventListener('scroll', scrollCollect)

  typeof preloader === 'object' && preloader.initLoading()
  document.getElementById('rightside').style.cssText = "opacity: ''; transform: ''"
  
  if (window.aplayers) {
    for (let i = 0; i < window.aplayers.length; i++) {
      if (!window.aplayers[i].options.fixed) {
        window.aplayers[i].destroy()
      }
    }
  }

  typeof typed === 'object' && typed.destroy()

  //reset readmode
  const $bodyClassList = document.body.classList
  $bodyClassList.contains('read-mode') && $bodyClassList.remove('read-mode')

  typeof disqusjs === 'object' && disqusjs.destroy()
})

document.addEventListener('pjax:complete', function () {
  window.refreshFn()

  document.querySelectorAll('script[data-pjax]').forEach(item => {
    const newScript = document.createElement('script')
    const content = item.text || item.textContent || item.innerHTML || ""
    Array.from(item.attributes).forEach(attr => newScript.setAttribute(attr.name, attr.value))
    newScript.appendChild(document.createTextNode(content))
    item.parentNode.replaceChild(newScript, item)
  })

  GLOBAL_CONFIG.islazyload && window.lazyLoadInstance.update()

  typeof chatBtnFn === 'function' && chatBtnFn()
  typeof panguInit === 'function' && panguInit()

  // google analytics
  typeof gtag === 'function' && gtag('config', '', {'page_path': window.location.pathname});

  // baidu analytics
  typeof _hmt === 'object' && _hmt.push(['_trackPageview',window.location.pathname]);

  typeof loadMeting === 'function' && document.getElementsByClassName('aplayer').length && loadMeting()

  // prismjs
  typeof Prism === 'object' && Prism.highlightAll()

  typeof preloader === 'object' && preloader.endLoading()
})

document.addEventListener('pjax:error', (e) => {
  if (e.request.status === 404) {
    pjax.loadUrl('/404.html')
  }
})</script><script async data-pjax src="//busuanzi.ibruce.info/busuanzi/2.3/busuanzi.pure.mini.js"></script></div></body></html>